Given the specific three-dimensional Cartesian coordinate point precisely defined algebraically as $P(-2, 2\sqrt{3}, 4)$, mathematically convert this exact point completely into both its cylindrical $(r, \theta, z)$ and spherical $(\rho, \theta, \phi)$ representations. Express all angles precisely in standard radians.

Determine exactly what specific three-dimensional physical shape is completely represented mathematically by the explicit spherical equation given as $\rho = 6 \sin \phi \cos \theta$. Convert the mathematical spherical equation completely into standard Cartesian $(x,y,z)$ coordinates precisely to verify your exact conclusion explicitly.